The chemical potential of a component in solution can be calculated from the chemical potential of the pure liquid, if the partial pressure above the solution is known.
When we derived the formula mi(p) = mi(0) + R*T*ln(p/p0) we assumed the system was behaving like an ideal gas. Assuming that the component is in the gas phase at standard conditions (the initial state), here we have two phases in the final state. As long as the pressure is below the vapor pressure p_vap and I consider the gas ideal, the formula is certainly valid as there is only the gas phase. The formula is still valid when the first drop of liquid appears at the vapor pressure. But for a single component system, as more and more liquid appears, the pressure stays constant at the value p_vap and we have different possible values for the molar volume. Are we saying that, for a single component system, the chemical potential is constant when the gas and liquid phases are in equilibrium? Its value seems to be the same from the moment the first drop of liquid appears to the moment the last bubble of gas disappears and the system is left with only the liquid phase.
You've got a misconception here. You're right that, for a single-component system, the pressure is constant at the vapor pressure when there is coexistence between the two phases -- regardless of whether there is only one drop of liquid or a large amount. And, consequently, as you point out, the chemical potential is the same value whenever the liquid and vapor are in coexistence. But, even though the volume of the liquid varies, the *molar* volume does not. The molar volume (inversely proportional to the density) has just a single value for this liquid under that pressure. So there is no inconsistency in the equations.
Because a condition of equality between gas and liquid/soln. is first imposed and the formula for u(soln.) later derived, does this means that a calculated u(soln.) for component A using this formula will only give us the value at equilibrium? What if the system is at disequilibrium with respect to A? - Thanks in advance, and for providing us with these videos. They are immensely helpful, and you a great lecturer.
Yes, that's true. This equation (and just about all of the thermodynamics in the course) is ONLY true at equilibrium. We typically can't provide algebraic equations for thermodynamic properties away from equilibrium. In this case, you could prepare a system with nearly arbitrary partial pressure above a solution with nearly arbitrary concentration, so there can't be an equation that relates one to the other. The best we can do in non-equilibrium systems is describe the *dynamics* for how the system approaches equilibrium. And even that is often very tricky.
I have used quite a few PChem textbooks, as you might imagine. For my own students, they have access to a draft textbook that I am working on. I also recommend the book by McQuarrie and Simon, which is quite good on most topics.
